Logistic regression 2

Average Error: 0.6 → 0.9

Time: 46.5s

Precision: binary64

Cost: 7748

Math

\[\log \left(1 + e^{x}\right) - x \cdot y \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1300000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot \frac{x}{4} - \left(0.5 \cdot x - \log 2\right)\right) - \frac{x}{-4}\right) - x \cdot y\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -1300000.0)
   (* y (- x))
   (- (- (- (* 3.0 (/ x 4.0)) (- (* 0.5 x) (log 2.0))) (/ x -4.0)) (* x y))))

C

double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}

↓

double code(double x, double y) {
	double tmp;
	if (x <= -1300000.0) {
		tmp = y * -x;
	} else {
		tmp = (((3.0 * (x / 4.0)) - ((0.5 * x) - log(2.0))) - (x / -4.0)) - (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = log((1.0d0 + exp(x))) - (x * y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1300000.0d0)) then
        tmp = y * -x
    else
        tmp = (((3.0d0 * (x / 4.0d0)) - ((0.5d0 * x) - log(2.0d0))) - (x / (-4.0d0))) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

↓

public static double code(double x, double y) {
	double tmp;
	if (x <= -1300000.0) {
		tmp = y * -x;
	} else {
		tmp = (((3.0 * (x / 4.0)) - ((0.5 * x) - Math.log(2.0))) - (x / -4.0)) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

↓

def code(x, y):
	tmp = 0
	if x <= -1300000.0:
		tmp = y * -x
	else:
		tmp = (((3.0 * (x / 4.0)) - ((0.5 * x) - math.log(2.0))) - (x / -4.0)) - (x * y)
	return tmp

Julia

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end

↓

function code(x, y)
	tmp = 0.0
	if (x <= -1300000.0)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(Float64(Float64(Float64(3.0 * Float64(x / 4.0)) - Float64(Float64(0.5 * x) - log(2.0))) - Float64(x / -4.0)) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1300000.0)
		tmp = y * -x;
	else
		tmp = (((3.0 * (x / 4.0)) - ((0.5 * x) - log(2.0))) - (x / -4.0)) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[x, -1300000.0], N[(y * (-x)), $MachinePrecision], N[(N[(N[(N[(3.0 * N[(x / 4.0), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 * x), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / -4.0), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.6
Target	0.1
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e6

   1. Initial program 0

      \[\log \left(1 + e^{x}\right) - x \cdot y \]

   2. Taylor expanded in x around 0 41.8

      \[\leadsto \color{blue}{\left(0.5 \cdot x + \log 2\right)} - x \cdot y \]

   3. Taylor expanded in x around inf 41.8

      \[\leadsto \color{blue}{0.5 \cdot x} - x \cdot y \]

   4. Taylor expanded in y around inf 0

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} \]

   5. Simplified 0

      \[\leadsto \color{blue}{y \cdot \left(-x\right)} \]
      Proof

      [Start] 0	\[ -1 \cdot \left(y \cdot x\right) \]
      rational_best-simplify-1 [<=] 0	\[ -1 \cdot \color{blue}{\left(x \cdot y\right)} \]
      rational_best-simplify-50 [=>] 0	\[ \color{blue}{y \cdot \left(x \cdot -1\right)} \]
      rational_best-simplify-11 [<=] 0	\[ y \cdot \color{blue}{\left(-x\right)} \]

   if -1.3e6 < x

   1. Initial program 0.7

      \[\log \left(1 + e^{x}\right) - x \cdot y \]

   2. Taylor expanded in x around 0 1.2

      \[\leadsto \color{blue}{\left(0.5 \cdot x + \log 2\right)} - x \cdot y \]

   3. Applied egg-rr 1.2

      \[\leadsto \color{blue}{\left(\left(3 \cdot \frac{x}{4} - \left(0.5 \cdot x - \log 2\right)\right) - \frac{x}{-4}\right)} - x \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1300000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot \frac{x}{4} - \left(0.5 \cdot x - \log 2\right)\right) - \frac{x}{-4}\right) - x \cdot y\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	13248

\[\log \left(1 + e^{x}\right) - x \cdot y \]

Alternative 2
Error	0.9
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq -1300000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log 2 + x\right) - 0.5 \cdot x\right) - x \cdot y\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq -1300000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x + \log 2\right) - x \cdot y\\ \end{array} \]

Alternative 4
Error	0.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + x\right) - x \cdot y\\ \end{array} \]

Alternative 5
Error	1.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -1300000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 6
Error	34.6
Cost	256

\[y \cdot \left(-x\right) \]

Alternative 7
Error	61.7
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023099 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))